Markov kernel

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In probability theory, a Markov kernel (or stochastic kernel) is a map that plays the role, in the general theory of Markov processes, that the transition matrix does in the theory of Markov processes with a finite state space.[1][2]

Formal definition

Let , be measurable spaces. A Markov kernel with source and target is a map with the following properties:

  1. The map is - measureable for every .
  2. The map is a probability measure on for every .

(i.e. It associates to each point a probability measure on such that, for every measurable set , the map is measurable with respect to the -algebra .)



describes the transition rule for the random walk on .

with i.i.d. random variables .



Semidirect product

Let be a probability space and a Markov kernel from to some . Then there exists a unique measure on , s.t.


Regular conditional distribution

Let be a Borel space, a - valued random variable on the measure space and a sub--algebra. Then there exists a Markov kernel from to , s.t. is a version of the conditional expectation for every , i.e.


It is called regular conditional distribution of given and is not uniquely defined.


The kernel can be estimated using kernel density estimation.[3]


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§36. Kernels and semigroups of kernels